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Methods of Solving Sequence and Series Problems

Ellina Grigorieva
Publication Date: 
Number of Pages: 
Problem Book
[Reviewed by
Allen Stenger
, on

The charm of this book is that it showcases a lot of problems that can be solved by ingenuity and elementary methods. It is a workbook rather than a textbook, and in terms of techniques does not go beyond what’s in most calculus courses: ratio and root tests, integral test, alternating series, and the sums of arithmetic and geometric progressions. It does a good job of addressing the common problem that students can quote the test but can only apply it if the series has exactly the form of that test. The problems have a strong Math Olympiad feel, but are drawn from many sources, including mathematical folklore. Most of the problems ask about a specific series or sequence rather than asking for proofs of general statements.

The writing is clear and there is a good narrative flow to keep you moving through what is essentially a series of worked problems. There are three chapters in the main exposition. There’s also a chapter of applications, all dealing with exponential growth or decay, so there’s not much breadth. There’s a final chapter labelled “Homework”; this is not greatly different in nature from the first three chapters, and includes hints and solutions; the problems may be a little less challenging than what went before. The production quality is good; the name Leibniz is misspelled at the top of p. 88, but I didn’t see any other errors.

A book with a somewhat similar approach, but that goes much deeper and is much harder, is Bonar and Khoury’s Real Infinite Series. Its source material is the Putnam Competitions and it is oriented more toward general theorems rather than specific series.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is His mathematical interests are number theory and classical analysis.

See the table of contents in the publisher's webpage.